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A213597 Triangle T(n,k), n>=1, 0<=k<=A000041(n), read by rows: row n gives the coefficients of the chromatic polynomial of the ranked poset L(n) of partitions of n, highest powers first.

%I A213597 #24 Feb 16 2025 08:33:17
%S A213597 1,0,1,-1,0,1,-2,1,0,1,-5,10,-9,3,0,1,-9,36,-79,98,-64,17,0,1,-17,136,
%T A213597 -666,2192,-5032,8111,-9013,6569,-2818,537,0,1,-28,378,-3242,19648,
%U A213597 -88676,306308,-819933,1703404,-2723374,3285552,-2887734,1739326,-639065,107435,0
%N A213597 Triangle T(n,k), n>=1, 0<=k<=A000041(n), read by rows: row n gives the coefficients of the chromatic polynomial of the ranked poset L(n) of partitions of n, highest powers first.
%C A213597 The ranked poset L(n) of partitions is defined in A002846.  A partition of n into k parts is connected to another partition of n into k+1 parts that results from splitting one part of the first partition into two parts.
%H A213597 Alois P. Heinz, <a href="/A213597/b213597.txt">Rows n = 1..9, flattened</a>
%H A213597 Olivier Gérard, <a href="/A002846/a002846.png">The ranked posets L(2),...,L(8)</a>
%H A213597 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>
%H A213597 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%e A213597 L(5):     (32)---(221)
%e A213597          /    \ /     \
%e A213597         /      X       \
%e A213597        /      / \       \
%e A213597     (5)---(41)---(311)---(2111)---(11111)
%e A213597 Chromatic polynomial: q^7-9*q^6+36*q^5-79*q^4+98*q^3-64*q^2+17*q.
%e A213597 Triangle T(n,k) begins:
%e A213597   1,   0;
%e A213597   1,  -1,   0;
%e A213597   1,  -2,   1,    0;
%e A213597   1,  -5,  10,   -9,    3,     0;
%e A213597   1,  -9,  36,  -79,   98,   -64,   17,     0;
%e A213597   1, -17, 136, -666, 2192, -5032, 8111, -9013, 6569, -2818, 537, 0;
%Y A213597 Row lengths give: 1+A000041(n) = A052810(n).
%Y A213597 Row sums (for n>1) and last elements of rows give: A000004.
%Y A213597 Columns k=1-2 give: A000012, (-1)*A000097(n-2).
%Y A213597 Cf. A002846, A213242, A213385, A213427.
%K A213597 sign,tabf
%O A213597 1,7
%A A213597 _Alois P. Heinz_, Jun 15 2012
%E A213597 Edited by _Alois P. Heinz_ at the suggestion of _Gus Wiseman_, May 02 2016